Desingularized propagating vortex equilibria

Core-Periphery Trading Networks

This study investigates the reinforcement correction factor of concrete core in more detail to prepare appropriate outlines for interpretation of results. This investigation aims to minimize uncertainties involved to carry out the more realistic condition assessment of suspect buildings before taking up retrofitting/strengthening measures. For this purpose, an extensive experimental program including different concrete strength level, moisture condition, core size, length-to-diameter (L/D) ratio of core and steel bar size with various configurations was under taken. The effect of variation of foregoing parameters on the ratio of average compressive strength of cores containing steel bars to that of corresponding plain specimens was probed. The results show that the correction factors are highly dependent on the values of volume percentage and position of reinforcing bars in the core, which are extremely interrelated. Hence, the experimental results do not show a good agreement with the provisions by the British Standard 6089 and Concrete Society Report No.11. Therefore, a statistical analysis on the prediction of cube compressive strength using linear and nonlinear regression models is accomplished. The results showed that regression models have great ability as a feasible tool for prediction of compressive strength on the basis of core testing in the presence of steel bars. However, the most accurate results obtained by nonlinear equations in comparison with linear models.

An expansion is developed of the two-point correlation function in the two-time correlation functions up to the expansion terms of third order. This expansion is necessary for the rigorous derivation of kinetic equations that objectively describe physical phenomena in a ‘single’ plasma with Langmuir turbulence. The correspondence of the expansion terms with various physical processes of the traditional weak plasma turbulence theory is discussed. Perspectives are outlined of application of the results to studies in the physics of non-ideal plasmas.

Singularly Perturbed Markov Chains with Two Small Parameters: A Matched Asymptotic Expansion

In this study, a singularly perturbed linear impulsive system with singularly perturbed impulses is considered. Many books discuss different types of singular perturbation problems. In the present work, an impulse system is considered in which a small parameter is introduced into the impulse equation. This is the main novelty of our study, since other works [25] have only considered a small parameter in the differential equation. A necessary condition is also established to prevent the impulse function from bloating as the parameter approaches zero. As a result, the notion of singularity for discontinuous dynamics is greatly extended. An asymptotic expansion of the solution of a singularly perturbed initial problem with an arbitrary degree of accuracy for a small parameter is constructed. A theorem for estimating the residual term of the asymptotic expansion is formulated, which estimates the difference between the exact solution and its approximation. The results extend those of [32], which formulates an analogue of Tikhonov’s limit transition theorem. The theoretical results are confirmed by a modelling example.

The channel coding problem in the moderate deviations regime is studied; here, the error probability sub-exponentially decays to zero, and the rate approaches the capacity slower than $O(1/\sqrt n )$. The main result refines Altuğ and Wagner’s moderate deviations result by deriving lower and upper bounds on the third-order term in the asymptotic expansion of the maximum achievable message set size. The third-order term of the expansion employs a new quantity called the channel skewness. For the binary symmetric channel and most practically important (n,ϵ) pairs, including n ∈ [100, 500] and ϵ ∈ [10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−10</sup> ,10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−1</sup> ], an approximation up to the channel skewness is the most accurate among several expansions in the literature.

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A study is carried out of the vacuum polarization in a strong Coulomb field. Radiative corrections are neglected. A perturbation calculation is avoided by making use of the explicit solutions of the Dirac equation in a Coulomb field. The Laplace transform of the polarization charge density times ${r}^{2}$ is found and used as a basis for further study. It is proved to be an analytic function of the strength of the inducing charge. It is verified that the first-order term in a power series expansion in the strength of the inducing charge just corresponds to the Uehling potential. The third-order term is studied in some detail. The leading term in the polarization potential close to the inducing charge and the space integral of the induced potential divided by $r$ are found to all orders in the strength of the inducing charge. Ambiguities are handled by a method corresponding to regularization.Some experimental applications are considered. The corrections to the Uehling term in these cases are found to be small.

For computation of hypersonic flowfields about space vehicles in low earth orbits, where the local Knudsen numbers (Kri) lie in the continuum-transition regime, a set of extended three-dimensional hydrodynamic equations are required which are more accurate than the NavierStokes equations and computationally more efficient than the Direct Simulation Monte Carlo (DSMC) computations. One such set is the Burnett equations which are obtained from the Chapman-Enskog expansion of the Boltzmann equation (with Knudsen number (Kri) as a small parameter) to O(Kn). In this paper, the threedimensional augmented Burnett equations are derived from the Chapman-Enskog expansion of the Boltzmann equation to O(Kn) and adding the augmented terms (linear third-order super Burnett terms with coefficients determined from linearized stability analysis to ensure stability of the augmented Burnett equations to small wavelength disturbances). The equations are solved using an explicit time-stepping scheme with Steger-Warming flux-vector splitting algorithm for the convective flux terms and second-order central differencing for the stress and heat flux terms. Maxwell-Smoluchowski slip boundary conditions are employed at the wall. 3-D augmented Burnett equations solver is developed both in FORTRAN?? and object-oriented JAVA. Both the FORTRAN and JAVA codes are applied to compute the three-dimensional hypersonic blunt-body flows for various range of Knudsen numbers and Mach numbers producing identical results. The focus of this paper is on development of a JAVA code for a very complex set of equations and its application to 3-D complex configurations. x Graduate Research Assistant * Bloomfield Distinguished Professor and Executive Director, Fellow AIAA Copyright © by authors. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. NOMENCLATURE et total energy per unit mass E total flux vector in x direction Ej convective flux vector in x direction EV viscous flux vector in x direction F total flux vector in y direction F! convective flux vector in y direction Fv viscous flux vector in y direction G total flux vector in z direction GI convective flux vector in z direction GV viscous flux vector in z direction Kn Knudsen number L reference length M Mach number Pr Prandtl number P pressure qt heat flux terms Q unknown vector in physical domain rn nose radius Re Reynolds number R gas constant T temperature Tw wall temperature u velocity components in x direction v velocity components in y direction w velocity components in z direction a accommodation coefficient for wall surface a/, P/ coefficients of stress terms in augmented Burnett equations y/ coefficients of heat flux terms in augmented Burnett equations K thermal conductivity K mean free path of gas molecules ILI viscosity 0, coefficients of heat flux terms in augmented Burnett equations p density a,y stress tensor a reflection coefficient for wall surface Y specific heat ratio co, coefficients of stress terms in augmented Burnett equations w circular frequency of perturbation American Institute of Aeronautics and Astronautics

The stability of the two-dimensional wake behind a circular cylinder - a free flow of general interest in differing applications (from aerodynamics to environmental physics and biology) - is studied by means of two different but complementary theoretical methods. The first part of the work is focused on the asymptotic evolution of disturbances described through modal analysis, a method which allows the determination of the asymptotic stability of a flow. The stability of the intermediate and far near-parallel wake is studied by means of a multiscale approach. The disturbance is defined as the local wavenumber at order zero in the longitudinal direction and is associated to a classical spatio-temporal WKBJ analysis. The inverse of the Reynolds number is taken as the small parameter for the multiscaling. It takes into account non-parallelism effects related to the transversal dynamics of the base flow. The first order corrections find absolute instability pockets in the first part of the intermediate wake (and not in the near wake, where the recirculating eddies are, as usually seen in literature in contrast with the near-parallelism hypothesis). These regions are present for Reynolds numbers larger than $Re=35$. That is in agreement with the general notion of critical Reynolds number for the onset of the first instability of about $Re=47$. In particular, for Re=50 and Re=100, the angular frequency obtained is in agreement with global data in literature concerning numerical and experimental results. The instability is convective throughout the domain. All the stability characteristics are vanishing in the far field, a fact that is independently confirmed by the asymptotic analysis of the Orr-Sommerfeld operator. Using asymptotic Navier-Stokes expansions for the wake inner field the entrainment evolution in the intermediate and far domain is evaluated in terms of asymptotic expansion. The maximum of entrainment is reached in the region where the absolute instability pockets are found. Downstream of this region the entrainment is decreasing and eventually vanishing in the far wake. This point confirms the validity of the multiscale approach. In the second part of the thesis the stability analysis is studied as an initial-value problem to observe the transient behaviour and the asymptotic state of perturbations initially imposed. The initial-value problem allows the formulation to be extended to the near-parallel flow configuration. The initial-value method is, however, less general than the modal analysis, since many parameters, such as the polar wavenumber, the spatial damping rate, the angle of obliquity and the symmetry of the perturbation, are involved. An exploratory analysis of these parameters permits the study of different transient configurations. Before the asymptotic (stable or unstable) state is reached, maxima and minima of the perturbation energy are observed for transients lasting hundreds of time scales. In the temporal asymptotics, the initial-value problem well reproduces modal results in terms of angular frequency and temporal growth rate. Moreover, for Reynolds numbers larger than the critical one (Re_{cr} = 47), the present method gives a good prediction, in terms of wavelength and pulsation, of the vortex shedding observed in experiments. In the framework of the initial-value problem formulation, a multiscale analysis for the stability of long waves is then proposed. Even to the lowest order, the multiscaling - whose small parameter is defined as the polar wavenumber - approximates sufficiently well the full problem solution with a relevant reduction of the computational cost. The two (modal and non-modal) analyses combined together lead to a quite complete description of the bluff-body wake stability

The Dirichlet problem for elliptic equations with a small parameter in the highest derivatives takes a unique place in mathematics. In general case it is impossible to build explicit solution to these problems, which is why the researchers apply different asymptotic methods. The aim of the research is to develop the asymptotic method of boundary functions for constructing complete asymptotic expansions of the solutions to such problems. The proposed generalized method of boundary functions differs from the matching method in the fact that the growing features of the outer expansion are actually removed from it and with the help of the auxiliary asymptotic series are fully included in the internal expansions, and differs from the classical method of boundary functions in the fact that the boundary functions decay in power-mode nature and not exponentially. Using the proposed method, a complete asymptotic expansion of the solution to the Dirichlet problem for bisingular perturbed linear inhomogeneous second-order elliptic equations with two independent variables in the ring with quadratic growth on the boundary is built. A built asymptotic series corresponds to the Puiseux series. The basic term of the asymptotic expansion of the solution has a negative fractional degree of the small parameter, which is typical for bisingular perturbed equations, or equations with turning points. The built expansion is justified by the maximum principle

On the enhanced drag force induced by permeation through a filtration membrane

A semi-analytical solution of the restricted two-body problem subject to an arbitrary low thrust is developed. The multiple scales perturbation method is used to split the problem into “slow” and “fast” scales. The small parameter that permits this approach is the ratio between thrust and gravitational acceleration. The thrust is expressed as a Fourier series expansion in terms of the fast scale, in which the coefficients are allowed to vary in the “slow” scale. The spectral analysis of the equations of motion gives rise to reduced dynamics of the “slow” motion. The results of the integration of this dynamical model are validated against the integration of the complete dynamics. Moreover, the preliminary results show that the propagation of the reduced dynamics are comparatively much more efficient than the complete dynamics, and show also t expected small errors of the method.

The topological derivative provides the sensitivity of a given shape functional with respect to an infinitesimal domain perturbation. Classically, this derivative comes from the second term of the topological asymptotic expansion, dealing only with infinitesimal perturbations. Therefore, as a natural extension of this concept, we can consider higher-order terms in the expansion. In particular, the next one we recognize as the second-order topological derivative, which allows us to deal with perturbations of finite sizes. This term depends explicitly on higher-order gradients of the solution associated to the non-perturbed problem and also implicitly through the solution of an auxiliary variational problem. In this article, we calculate the explicit as well as implicit terms of the second-order topological asymptotic expansion for the total potential energy associated to the Laplace equation in the two-dimensional domain. The domain perturbation is done by the insertion of a small inclusion with a thermal conductivity coefficent value different from the bulk material. Finally, we present some numerical experiments showing the influence of the second-order term in the topological asymptotic expansion for several values of the thermal conductivity coefficent of the inclusion.

In this study, we demonstrate, for the first time, the existence of a short-wave instability in a Lamb–Oseen vortex subjected to a triangular strain field generated by three satellite vortices, which we term the triangular instability. We identify this instability by numerically integrating the linearised Navier–Stokes equations around a quasi-steady base flow to capture the most unstable mode and validate it by comparing results with theoretical predictions. We evaluate this instability by calculating the growth rates associated with the parametric resonant coupling of two Kelvin waves with the triangular strain field in the limit of small strain rate and large Reynolds number. Our analysis reveals that resonance occurs only for combinations of the azimuthal wavenumbers $m = 1$ and $m = - 2$ (or their symmetric counterparts with opposite signs). We observe several unstable modes with positive growth rates for a moderate viscous Reynolds number $10^4$ and straining parameter value $\epsilon = 0.008$ , defined as the cube of the ratio of the core size to the distance from the satellite vortices. The most unstable mode, dominant at typically high Reynolds numbers, has $k \approx 5.18/a$ and $\omega \approx - 0.312\Omega$ (where $a$ and $\Omega$ denote the core size and central angular velocity). It exhibits negligible critical layer damping and remains the most unstable mode over a wide range of ${Re}$ and $\epsilon$ . At lower Reynolds numbers, another mode with $k \approx 1.76/a$ and $\omega \approx - 0.407\Omega$ , despite significant critical layer damping, becomes the most unstable.